A canoe of mass 38 kg lies at rest in still water. A man and a woman are at opposite ends of the canoe 4.0 m apart and symmetrically located with respect to the canoe’s centre (which is also its centre of mass). The mass of the man is 65 kg and the woman’s mass is smaller.

The two people then change places and the man observes that the canoe shifts a distance 0.20 m relative to the water. What is the woman’s mass?

To solve this problem, we can use the principle of conservation of momentum. When the people change places, there is no external force acting on the system, so the total momentum before the change is equal to the total momentum after the change.

Before the change, the momentum of the system is zero because the canoe and its occupants are at rest. After the change, the canoe shifts by a distance relative to the water. This shift indicates that momentum has been transferred to the canoe from the people.

To calculate the woman's mass, we need to determine the momentum transferred to the canoe. The momentum can be calculated using the equation:

Momentum = mass * velocity

Initially, the man's momentum is given by:

Momentum_man_initial = mass_man * velocity_man_initial

Since the canoe is at rest, the initial velocity of the man relative to the water is zero.

Therefore, Momentum_man_initial = mass_man * 0 = 0

When the people change places, the canoe shifts. This shift indicates that momentum has been transferred to the canoe, causing it to move relative to the water. Therefore, the final momentum of the system is:

Momentum_final = mass_man * velocity_man_final + mass_woman * velocity_woman_final

Because the momentum change is transferred to the canoe and given by:

Momentum_change = mass_woman * velocity_woman_final

We can rewrite the equation for momentum_final as:

Momentum_final = Momentum_man_initial + Momentum_change

Since the initial momentum of the man is zero, we have:

Momentum_final = 0 + Momentum_change
Momentum_final = Momentum_change

We know that the canoe shifts a distance of 0.20 m relative to the water. Since the man and the woman are 4.0 m apart, and they change places symmetrically, the woman moves a distance of:

Distance_woman = 0.20 m / 4.0 m = 0.05 m

To find the woman's velocity, we can use the equation:

Velocity_woman_final = distance_woman / time

We don't have the exact time it takes for the woman to move, but we can assume it's the same for both the man and the woman. Therefore, we can ignore the time in this calculation.

Finally, we can substitute the values into the equation to find the woman's mass:

Momentum_change = mass_woman * velocity_woman_final

0.20 m = mass_woman * (0.05 m / time)

Simplifying the equation:

mass_woman = momentum_change / velocity_woman_final
mass_woman = 0.20 m / (0.05 m / time)

The exact mass of the woman cannot be determined without knowing the time it takes for them to change places. You will need additional information such as the time or the velocity of the woman relative to the water to calculate the woman's mass.